note09 - Chapter 8 Nonparametric Bootstrap Methods 1 The...

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Unformatted text preview: Chapter 8: Nonparametric Bootstrap Methods 1. The Basic Bootstrap Method • to measure how close the statistical estimate is to the population parameter • for the population mean μ – the approximate 95% CI is given by ¯ X ± 2 S X √ n . – the margin of error ( 2 S X n- 1 / 2 ) : a measure of how accurately ¯ X esti- mates μ • to present a simple method for obtaining a margin of error for problems in which analytical solutions are not available ⇒ the method is called the bootstrap (a) Mean Squared Error and Margin of Error • the mean square error (MSE) of the estimate: MSE = E parenleftBig ˆ θ- θ parenrightBig 2 where θ is a population parameter and ˆ θ is a statistical estimate • the Chebyshev-Markov inequality: Pr parenleftBig | ˆ θ- θ | ≤ k √ MSE parenrightBig ≥ 1- 1 k 2 – in the case of estimating μ , the probability may be much higher – the quantity k √ MSE : the margin of error of the estimate – the margin of error can be used as a rough measure of the accuracy of an estimate when other measures may not be readily available • when an explicit formula for the MSE of an estimate is not available ˆ MSE = 1 T T summationdisplay t =1 parenleftBig ˆ θ t- θ parenrightBig 2 where ˆ θ t is the estimate of θ from the t th repetition of the sampling and T is the number of times the sampling experiment is repeated 1 (b) The Bootstrap Estimate of MSE • the population distribution is generally not known: the data may be used as a substitute for the population ⇒ resample the data • to simulate sampling from an infinite population, sampling is done with replacement ⇒ a bootstrap sample • The steps for obtaining a bootstrap estimate of MSE: i. ˆ θ from the original data ii. Take T independent bootstrap samples x * 1 , x * 2 , . . ., x * T , each con- sisting of n data values drawn with replacement from x . Typically, ≥ 1000 iii. Compute ˆ θ b,t : the estimate of θ from x * t iv. Obtain the bootstrap MSE as ˆ MSE = 1 T T summationdisplay t =1 parenleftBig ˆ θ b,t- ˆ θ parenrightBig 2 • bootstrap variance and bias – the bias B : B = E parenleftBig ˆ θ parenrightBig- θ MSE = V ar parenleftBig ˆ θ parenrightBig + B 2 – in place of step iv. ˆ E = 1 T T summationdisplay t =1 ˆ θ b,t ˆ B = ˆ E- ˆ θ V ar parenleftBig ˆ θ parenrightBig = 1 T T summationdisplay t =1 parenleftBig ˆ θ b,t- ˆ E parenrightBig 2 2 • number of bootstrap samples – Efron and Tibshirani (1993): if 50 ≤ T ≤ 200, ˆ se boots a good estimator of the standard error – Booth and Sarkar (1998): T ≥ 800 – if not too much burden for computation, T ≥ 1000 safe • nonparametric vs parametric bootstrap estimates – nonparametric bootstrap: no assumptions are made about the functional form of the population distribution – parametric bootstrap: assumptions are made about the form of the population distribution example with a normal distribution i. compute ˆ θ , ¯ x and S 2 x from data x ii. Takeii....
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note09 - Chapter 8 Nonparametric Bootstrap Methods 1 The...

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